Whether the signal lives on an algebraic variety — a surface defined by polynomial equations.
Delay-embeds the signal and tests whether the resulting point cloud lies near the zero set of a low-degree polynomial. In algebraic geometry, the Zariski topology's closed sets are exactly these zero sets, making the topology non-Hausdorff (most pairs of points can't be separated by open sets). This geometry also measures the signal's pattern lattice: does it obey Boolean logic (classical), or does it deviate toward a Heyting algebra (intuitionistic)?
How far is the signal's pattern lattice from Boolean? 1.0 means maximally non-Boolean: the law of excluded middle fails for many pattern pairs. Forest fire scores 1.0 — its intermittent dynamics create pattern relationships that violate classical logic (a pattern can be "neither always present nor always absent" in a meaningful sense). Logistic period-3 (0.90) and period-5 (0.78) are high too: their windows of periodicity within chaos create ambiguous pattern membership. Logistic full chaos and Rössler score 0.0 — fully chaotic systems are "Boolean" because every pattern either appears or doesn't, with no intermediate states.
What fraction of point pairs are non-separable in the Zariski topology? Collatz gap lengths score 0.9995 (almost all pairs are non-separable — the data lies almost exactly on an algebraic variety). Rainfall scores 0.96 (its exponential-like distribution creates a low-dimensional algebraic structure). Wichmann-Hill (0.007) is nearly zero — good PRNGs fill delay space uniformly, avoiding any algebraic surface.
How well does a low-degree polynomial fit the delay-embedded cloud? Dice rolls (0.064) have the highest residual — the data is maximally far from any algebraic variety. Constants score 0.0 (trivially on a variety: the point {(c,c,c,...)}). This is the "anti-algebraic" metric: high values mean the signal has no polynomial recurrence relation.
Slope of log-residual vs polynomial degree. Sawtooth (-5.21) and L-System Dragon (-4.40) have the steepest negative slopes — residuals drop rapidly with degree, meaning a low-degree polynomial captures the structure. Constants score ~0 (no slope to measure). Strongly negative slopes indicate algebraic recurrence relations; near-zero slopes mean the data resists polynomial fitting at all degrees.
Second derivative (curvature) of the log-residual curve across degrees 1-4. Collatz Parity (14.5), Rule 30 (14.4), and Symbolic Lorenz (14.3) score highest — their residual curves have strong convexity, meaning the rate of improvement accelerates at higher degrees. Rössler and constants score 0.0. High convexity signals that the data has algebraic structure at a specific degree that lower degrees miss.
Consistency of the heyting_gap across non-overlapping data segments. Logistic period-3 (90.5) scores highest among non-degenerate sources — its intermittent dynamics produce consistently non-Boolean pattern structure throughout the signal. Rössler and logistic period-2 score 0.0 (their heyting_gap is either always zero or varies chaotically between segments). High stability means the non-Hausdorff property is a genuine structural feature, not a statistical fluctuation.
| Source | Domain | Value |
|---|---|---|
| Moebius Function | number_theory | 0.0716 |
| Partition Function | number_theory | 0.0687 |
| Dice Rolls | exotic | 0.0640 |
| ··· | ||
| Pell Word | exotic | 0.0000 |
| Logistic r=3.2 (Period-2) | chaos | 0.0000 |
| Collatz Parity | number_theory | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Forest Fire | exotic | 1.0000 |
| Pell Word | exotic | 0.9048 |
| Logistic r=3.83 (Period-3 Window) | chaos | 0.9048 |
| ··· | ||
| Prime Gaps | number_theory | 0.0000 |
| Rossler Attractor | chaos | 0.0000 |
| Gzip (level 1) | binary | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Tribonacci Word | exotic | 1.0000 |
| Pell Word | exotic | 1.0000 |
| Fibonacci Word | exotic | 1.0000 |
| ··· | ||
| Lorenz Attractor | chaos | 0.0000 |
| Rossler Attractor | chaos | 0.0000 |
| Sawtooth Wave | waveform | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Mian-Chowla | number_theory | 0.9821 |
| Aubry-André Critical | quantum | 0.9767 |
| Rainfall (ORD Hourly) | climate | 0.9601 |
| ··· | ||
| Arnold Cat Map | chaos | 0.0065 |
| √2 Digits | number_theory | 0.0065 |
| Euler-Mascheroni γ Digits | number_theory | 0.0066 |
| Source | Domain | Value |
|---|---|---|
| Pell Word | exotic | 14.5042 |
| Collatz Parity | number_theory | 14.4790 |
| Prime Indicator | number_theory | 14.4375 |
| ··· | ||
| Rossler Attractor | chaos | 0.0000 |
| Gzip (level 1) | binary | 0.0000 |
| Gzip (level 9) | binary | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | -0.0406 |
| Human Proteome | bio | -0.6883 |
| Dice Rolls | exotic | -0.6995 |
| ··· | ||
| Moebius Function | number_theory | -5.7252 |
| Padovan Word | exotic | -5.6885 |
| Bernoulli Shift | chaos | -5.5469 |
Zariski's heyting_gap (CV=1.76) is the framework's most variable metric across sources. It detects a specific structural property — intermittency — that other geometries approximate but don't directly measure. The forest fire / logistic period-3 cluster at the top of the heyting_gap ranking corresponds to "intermittent chaos": systems that switch between qualitatively different dynamical regimes. In investigations, heyting_gap discriminated between normal and abnormal ECG with moderate effect size, because cardiac arrhythmias are fundamentally intermittent.